Physics
First Order Theory
First Order Theory is a mathematical framework used to describe physical systems. It involves the use of first-order logic, which is a formal system for reasoning about statements. In physics, this theory is often used to describe the behavior of particles and their interactions.
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No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 4 First-Order Logic First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus , quantification theory , and predicate logic. First-order logic is distinguished from propositional logic by its use of quantifiers; each interpretation of first-order logic includes a domain of discourse over which the quantifiers range. Briefly, first-order logic is distinguished from higher-order logics in that quantification is allowed only over atomic entities (individuals but not sets). There are many deductive systems for first-order logic that are sound (only deriving correct results) and complete (able to derive any logically valid implication). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is of great importance to the foundations of mathematics, where it has become the standard formal logic for axiomatic systems. It has sufficient expressive power to formalize two important mathematical theories: Zermelo–Fraenkel set theory (ZF) and first-order Peano arithmetic. However, no axiom system in first order logic is strong enough to fully (categorically) describe infinite structures such as the natural numbers or the real line. Categorical axiom systems for these structures can be obtained in stronger logics such as second-order logic. A history of first-order logic and an account of its emergence over other formal logics is provided by Ferreirós (2001).
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 8 First-Order Logic and Frege's Propositional Calculus First-order logic First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus , quantification theory , and predicate logic. First-order logic is distinguished from propositional logic by its use of quantifiers; each inter-pretation of first-order logic includes a domain of discourse over which the quantifiers range. Briefly, first-order logic is distinguished from higher-order logics in that quantification is allowed only over atomic entities (individuals but not sets). There are many deductive systems for first-order logic that are sound (only deriving correct results) and complete (able to derive any logically valid implication). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is of great importance to the foundations of mathematics, where it has become the standard formal logic for axiomatic systems. It has sufficient expressive power to formalize two important mathematical theories: Zermelo–Fraenkel set theory (ZF) and first-order Peano arithmetic. However, no axiom system in first order logic is strong enough to fully (categorically) describe infinite structures such as the natural numbers or the real line. Categorical axiom systems for these structures can be obtained in stronger logics such as second-order logic. A history of first-order logic and an account of its emergence over other formal logics is provided by Ferreirós (2001).
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 5 First-Order Logic and Frege's Propositional Calculus First-order logic First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus , quantification theory , and predicate logic. First-order logic is distinguished from propositional logic by its use of quantifiers; each inter-pretation of first-order logic includes a domain of discourse over which the quantifiers range. Briefly, first-order logic is distinguished from higher-order logics in that quantification is allowed only over atomic entities (individuals but not sets). There are many deductive systems for first-order logic that are sound (only deriving correct results) and complete (able to derive any logically valid implication). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is of great importance to the foundations of mathematics, where it has become the standard formal logic for axiomatic systems. It has sufficient expressive power to formalize two important mathematical theories: Zermelo–Fraenkel set theory (ZF) and first-order Peano arithmetic. However, no axiom system in first order logic is strong enough to fully (categorically) describe infinite structures such as the natural numbers or the real line. Categorical axiom systems for these structures can be obtained in stronger logics such as second-order logic. A history of first-order logic and an account of its emergence over other formal logics is provided by Ferreirós (2001).
eBook - PDF- Neville Dean(Author)
- 2017(Publication Date)
- Red Globe Press(Publisher)
A body of knowledge consists of all the true statements that can be made about the area concerned. For example, the theory of integer arithmetic consists of all the true statements that can made about whole numbers. Such statements include: 1 + 1 = 2 45 2 − 2 × ( − 78 ) = 2181 ∀ x ∃ y y > x 736 / 32 = 23 √ 64 = 8 A theory can be about a specific entity such as a network. Alternatively it can be about a collection of entities, in which case the theory describes the general properties shared by all such entities; thus network theory is the body of knowledge which applies to all networks. All theories are infinite, that is there are infinitely many true statements we can make about anything. The reason for this is simple. Any statement which is an instantiation of a tautology must be true by definition. But there are infin-itely many tautologies. Hence there must be infinitely many true statements. The outcome of this is that it is not possible in practice to store a complete theory in a computer system, or in a book. Now, we have seen how we can use logic to obtain new items of information from existing ones by means of de-duction. We can use this to reduce the number of items of information we need to hold, and to regenerate the missing items by logic. A common approach is to choose a set of axiom s, which are basic truths of the theory, and to prove other results using first order logic. We say that the axioms define a First Order Theory . Ideally the First Order Theory will not give rise to any statement which is false, that is the First Order Theory will be sound. Furthermore, we also want to be able to prove all true results using the First Order Theory, that is we want the First Order Theory to be complete. Unfortunately, first order theories are not always sound and complete; in particular, it can be shown that it is im-possible for a First Order Theory of theory of arithmetic to be both sound and complete.
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